On a conjecture of Herzer concerning linear partitions
type: Item
instance of: scholarly article
title: On a conjecture of Herzer concerning linear partitions
zbMATH Open document ID: 0611.20018
DOI: 10.1007/BF01227842
author: Ralph-Hardo Schulz
published in: Journal of Geometry
publication date: 1986-01-01T00:00:00Z
review text: A partition \(\pi\) (G) of a finite group G is a set of non-identity subgroups such that each non-identity element of G lies in exactly one of the subgroups. A partition is nontrivial if it contains at least two subgroups and is called linear if there is a nonempty set \(\epsilon\) (G) of subgroups of G such that (i) each element of \(\epsilon\) (G) is the union of at least two different elements of \(\pi\) (G), (ii) any two different elements of \(\pi\) (G) are contained in a unique element of \(\epsilon\) (G), (iii) \(G\not\in \epsilon (G).\) The author answers a conjecture of A. Herzer by proving that a finite group with a linear partition is either an abelian p-group, or a Frobenius group. The proof depends on the classification of finite groups with non-trivial partitions.
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 3987552
zbMATH Keywords: finite group
zbMATH Keywords: linear partition
zbMATH Keywords: abelian p-group
zbMATH Keywords: Frobenius group
zbMATH Keywords: partitions
MaRDI profile type: Publication
Wikidata QID: Q122917271
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description: scientific article; zbMATH DE number 3987552